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The Circumsphere Radius of a Truncated Icosahedron is the radius of the sphere that contains the polyhedron such that all its vertices lie on the surface of the sphere. The truncated icosahedron is an Archimedean solid with 12 regular pentagonal faces and 20 regular hexagonal faces.
The calculator uses the formula:
Where:
Explanation: This formula calculates the radius of the sphere that circumscribes a truncated icosahedron based on its original icosahedral edge length before truncation.
Details: Calculating the circumsphere radius is important in geometry, material science, and architecture for understanding the spatial properties and packaging efficiency of this particular polyhedral structure.
Tips: Enter the icosahedral edge length in meters. The value must be positive and greater than zero.
Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid obtained by truncating the vertices of a regular icosahedron, resulting in 12 pentagonal and 20 hexagonal faces.
Q2: Where is the truncated icosahedron commonly found?
A: The truncated icosahedron is best known as the shape of a soccer ball and is also found in carbon fullerene molecules (C60 buckyballs).
Q3: What are the units for measurement?
A: The calculator uses meters for both input and output, but the formula works with any consistent unit system.
Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the truncated icosahedron and its relationship to the original icosahedron.
Q5: What is the significance of the constants in the formula?
A: The constants \(\sqrt{58 + 18\sqrt{5}}\) and the division by 12 are derived from the geometric properties and trigonometric relationships within the truncated icosahedron structure.